James Phillips's Lecture Notes, [18--]: Electronic Edition.Phillips, James, 1792-1867Funding from the University Library, University of North Carolina at Chapel Hill supported the electronic publication of this title.Text transcribed byBari HelmsImages scanned byBari HelmsText encoded byBrian DietzFirst Edition, 2005ca. 10KThe University Library, University of North Carolina at Chapel Hill Chapel Hill, North Carolina2005

2. Let us consider a function in its state of augmentation, in consequence of the increase of the variable which it contains. As every function of a variable x can be represented by the ordinate of a curve BMM', let AP = x and PM = y be the ordinates of a point M of this curve, and let us suppose that the abscissa AP receives an increment PP' = h; the ordinate PM will become P'M' = y'. Fig. 1. To obtain the value of this new ordinate, we see that it is necessary to change x into x + h, in the equation of the curve, and the value which this equation will then determine for y will that of y'.

For example, if we had the equation y = mx2, we should obtain y' by changing x into x + h, and y into y', and we would have

Let us see what this result teaches us: y' - y represents the increment of the function y in consequence of the increment h given to x, since this difference y' - y is that of the new state of magnitude of y, as respects its primitive state.

On the other hand the increment of x being h, it follows that [y'/h - y/h] is the ratio of the increment [...]